Due to Lemma REF , it is easily seen that every Loewner function is also a Herglotz-Nevanlinna function. The converse result is dependant on the number of variables \(n\) . When \(n=1\) , it follows from the results summarized in the beginning of Section  that every Herglotz-Nevanlinna function is also a Loewner function. If \(n=2\) , this still holds, with the result being a consequence of a theorem concerning Schur function on the polydisk, see [1]} and [2]}. When \(n \ge 3\) , Loewner functions form a proper subclass of Herglotz-Nevanlinna functions, with the result being a consequence of the theory of commuting contractions [3]}, [4]}.
